Maximum power tracking technique for solar panels

ABSTRACT

The present invention provides an apparatus and method for tracking the maximum power point of a solar panel. A pulsewidth-modulated converter, for example a SEPIC or Cuk converter, is provided between the output of the panel and the load, and a perturbation is introduced into a switching parameter of the converter.

FIELD OF THE INVENTION

[0001] This invention relates to method and apparatus for efficientlyextracting the maximum output power from a solar panel under varyingmeteorological and load conditions.

BACKGROUND OF THE INVENTION

[0002] The solar panel is the fundamental energy conversion component ofphotovoltaic (PV) systems which have been used in many applications,such as the aerospace industry, electric vehicles, communicationequipment, and others. As solar panels are relatively expensive, it isimportant to improve the utilization of solar energy by solar panels andto increase the efficiency of PV systems. Physically, the power suppliedby the panels depends on many extrinsic factors, such as insolation(incident solar radiation) levels, temperature, and load condition.Thus, a solar panel is typically rated at an insolation level togetherwith a specified temperature, such as 1000W/m² at 25° C. The electricalpower output of a solar panel usually increases linearly with theinsolation and decreases with the cell/ambient temperature.

PRIOR ART

[0003] In practice, there are three possible approaches for maximizingthe solar power extraction in medium- and large-scale PV systems. Theyare sun tracking, maximum power point (MPP) tracking or both. For thesmall-scale systems, the use of MPP tracking only is popular for theeconomical reason. In the last two decades, various methods includingpower-matching schemes, curve-fitting techniques, perturb-and-observemethods, and incremental conductance algorithms have been proposed fortracking the MPP of solar panels.

[0004] Power-matching schemes require the selected solar panels to havesuitable output characteristics or configurations that can be matchedwith particular loads. However, these techniques only approximate thelocation of the MPP because they are basically associated with specificinsolation and load conditions. Curve-fitting techniques require priorexamination of the solar panel characteristics, so that an explicitmathematical function describing the output characteristics can bepredetermined. Proposed prior methods are based on fitting the operatingcharacteristic of the panel to the loci of the MPP of the PV systems.Although these techniques attempt to track the MPP without computing thevoltage-current product explicitly for the panel power, curve-fittingtechniques cannot predict the characteristics including other complexfactors, such as aging, temperature, and a possible breakdown ofindividual cells.

[0005] The perturb-and-observe (PAO) method is an iterative approachthat perturbs the operation point of the PV system, in order to find thedirection of change for maximizing the power. This is achieved byperiodically perturbing the panel terminal voltage and comparing the PVoutput power with that of the previous perturbation cycle. Maximum powercontrol is achieved by forcing the derivative of the power to be equalto zero under power feedback control. This has an advantage of notrequiring the solar panel characteristics. However, this approach isunsuitable for applications in rapidly changing atmospheric conditions.The solar panel power is measured by multiplying its voltage andcurrent, either with a microprocessor or with an analog multiplier. Incertain prior methods, the tracking technique is based on the fact thatthe terminal voltage of the solar panels at MPP is approximately at 76%of the open-circuit voltage, but this means that in order to locate theMPP, the panel is disconnected from the load momentarily so that theopen-circuit voltage can be sampled and kept as reference for thecontrol loop.

[0006] The disadvantages of the PAO method can be mitigated by comparingthe instantaneous panel conductance with the incremental panelconductance. This method is the most accurate one among the above priorart methods and is usually named as the incremental conductancetechnique (ICT). The input impedance of a switching converter isadjusted to a value that can match the optimum impedance of theconnected PV panel.

[0007] This technique gives a good performance under rapidly changingconditions. However, the implementation is usually associated with amicrocomputer or digital signal processor that usually increases thewhole system cost.

SUMMARY OF THE INVENTION

[0008] According to the present invention there is provided a method fortracking the maximum power point of a solar panel, comprising:

[0009] (a) providing a pulsewidth-modulated (PWM) DC/DC converterbetween the output of said panel and a load, and

[0010] (b) introducing a perturbation into a switching parameter of saidconverter.

[0011] In a first embodiment of the invention the parameter is the dutycycle of at least one switching device in the converter. In a secondembodiment of the invention the parameter is the switching frequency ofat least one switching device in the converter.

[0012] According to another aspect of the invention there is providedapparatus for tracking the maximum power point of a solar panel,comprising:

[0013] (a) a pulsewidth-modulated (PWM) DC/DC converter between theoutput of the solar panel and a load, and

[0014] (b) means for introducing a perturbation into a switchingparameter of said converter.

[0015] In the first embodiment of the invention the converter operatesin switching mode and said perturbation means comprises means forintroducing a perturbation into the duty cycle of at least one switchingdevice in the said converter. In a second embodiment of the inventionthe converter operates in switching mode and said perturbation meanscomprises means for introducing a perturbation into the switchingfrequency of at least one switching device in the said converter.

BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Some examples of the present invention will now be described byway of example and with reference to the accompanying drawings, inwhich:

[0017]FIG. 1 is an equivalent circuit of a solar-panel connected to aconverter,

[0018]FIG. 2 is a circuit diagram of a SEPIC converter,

[0019]FIG. 3 illustrates the operating principles of a SEPIC converter,

[0020]FIG. 4 is a block diagram of a first embodiment of the invention,

[0021]FIG. 5 illustrates an experimental set-up,

[0022] FIGS. 6(a) & (b) illustrate solar panel characteristics in thefirst embodiment,

[0023] FIGS. 7(a) & (b) show converter waveforms in the firstembodiment,

[0024] FIGS.. 8(a) & (b) show further converter waveforms in the firstembodiment,

[0025] FIGS.9(a) & (b) show further converter waveforms in the firstembodiment,

[0026]FIG. 10 shows further converter waveforms in the first embodiment,

[0027]FIG. 11 is a comparison of maximum solar panel output power usingthe first embodiment with ideal power output,

[0028]FIG. 12 is a circuit diagram of a Cuk converter,

[0029]FIG. 13 illustrates the relationship between ε₁/β and k,

[0030]FIG. 14 is a block diagram of a method and apparatus for MPPtracking according to a second embodiment of the invention,

[0031]FIG. 15 illustrates the relationship between ε₂/β and k′,

[0032]FIG. 16 illustrates an experimental set up,

[0033]FIG. 17 shows the performance of a solar panel with MPP trackingaccording to the second embodiment of the invention,

[0034] FIGS.18(a) and (b) show converter waveforms in the secondembodiment of the invention with the converter in DICM and DCVM modesrespectively,

[0035] FIGS. 19(a)-(d) show converter waveforms in the second embodimentof the invention with the converter in DICM ((a) and (c)) and DCVM ((b)and (d)) modes respectively,

[0036] FIGS. 20(a) and (b) show further converter waveforms in thesecond embodiment with the converter in DICM and DCVM modesrespectively, and

[0037] FIGS. 21(a) and (b) show further converter waveforms in thesecond embodiment with the converter in DICM and DCVM modesrespectively.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

[0038] Before describing a first embodiment of the invention in detail,a theoretical explanation of the principles underlying the presentinvention is provided.

[0039] A. Derivation of the Required Dynamic Input Characteristics of aConverter at MPP

[0040]FIG. 1 shows an equivalent circuit of the solar panel connected toa converter. The solar panel is represented by a voltage source v_(g)connected in series with an output resistance r_(g) at the MPP. Theinput voltage and the equivalent input resistance of the converter arev_(i) and r_(i), respectively. As the input power P_(i) to the converteris equal to the output power P_(o) of the solar panel, $\begin{matrix}{P_{i} = {P_{o} = \frac{v_{i}^{2}}{r_{i}}}} & (1)\end{matrix}$

[0041] The rate of change of P_(i) with respect to v_(i) and r_(i) canbe shown to be $\begin{matrix}{{\partial P_{i}} = {{2\frac{v_{i}}{r_{i}}{\partial v_{i}}} - {\frac{v_{i}^{2}}{r_{i}^{2}}{\partial r_{i}}}}} & (2)\end{matrix}$

[0042] At the MPP, the rate of change of P_(i) equals zero. Hence,$\begin{matrix}{{\partial P_{i}} = {\left. 0\Rightarrow\frac{\partial v_{i}}{\partial r_{i}} \right. = \frac{V_{i}}{2R_{i}}}} & (3)\end{matrix}$

[0043] where V_(i) and R_(i) are the input voltage and the inputresistance at MPP.

[0044] The above equation gives the required dynamic inputcharacteristics of the converter at the MPP. The input voltage will havea small-signal variation of δv _(i). the input resistance is subject toa small-signal change of δr _(i). That is, $\begin{matrix}{{\frac{\delta \quad v_{i}}{\delta \quad r_{i}} \approx \frac{\partial v_{i}}{\partial r_{i}}} = \frac{V_{i}}{2R_{i}}} & (4)\end{matrix}$

[0045] In the following sections, a SEPIC converter is illustrated. Itwill be understood, however, that similar techniques can be applied toother converters, such as Cuk, buck-boost, buck, and boost converters.

[0046] B. Input Resistance and Voltage Stress of a SEPIC Converter

[0047]FIG. 2 shows the circuit diagram of a SEPIC converter. If theconverter is operated in discontinuous capacitor voltage (DCV) mode,there are in total three circuit topologies in one switching cycle (d).The sequence of operation and the waveforms are shown in FIG. 3. If thetwo inductor currents (i.e., I₁ and I₂) are assumed to be constant, thecapacitor voltage v_(c)(t) and diode voltage v_(D)(t) in the respectivethree operating intervals can be expressed as $\begin{matrix}{{v_{C}(t)} = \left\{ \begin{matrix}{\frac{{I_{1}\left( {1 - d} \right)}T_{S}}{C} - V_{o} - {\frac{I_{2}}{C}t}} & {0 < t < {d_{1}T_{S}}} \\{- V_{o}} & {{d_{1}T_{S}} < t < {d\quad T_{S}}} \\{{\frac{I_{1}}{C}\left( {t - {d\quad T_{S}}} \right)} - V_{o}} & {{d\quad T_{S}} < t < T_{S}}\end{matrix} \right.} & \text{(5a)} \\{{v_{D}(t)} = \left\{ {{{\begin{matrix}{V_{o} + {v_{C}(t)}} & {0 < t < {d_{1}T_{S}}} \\0 & {{d_{1}T_{S}} < t < T_{S}}\end{matrix}\text{As}\quad {v_{C}\left( {d_{1}T_{S}} \right)}} = {- V_{o}}},} \right.} & \text{(5b)} \\\begin{matrix}{{\frac{{I_{1}\left( {1 - d} \right)}T_{S}}{C} - V_{o} - {\frac{I_{2}}{C}d_{1}T_{S}}} = \quad {- V_{o}}} \\{\left. \Rightarrow\quad d_{1} \right. = \quad {\frac{I_{1}}{I_{2}}\left( {1 - d} \right)}}\end{matrix} & (6)\end{matrix}$

[0048] Under the steady-state condition, the average voltage across L₂is zero. Hence, the V_(o) is equal to the average value of v_(D). Thatis, $\begin{matrix}{V_{o} = {{\frac{1}{T_{S}}{\int_{0}^{D_{1}T_{S}}{{v_{D}(t)}{t}}}} = {\frac{T_{S}}{2C}{I_{1}\left( {1 - d} \right)}d_{1}}}} & (7)\end{matrix}$

[0049] As the average voltage across L₁ is also zero, $\begin{matrix}{v_{i} = {{\frac{1}{T_{S}}{\int_{0}^{T_{S}}{{v_{C}(t)}{t}}}} = {\frac{T_{S}}{2C}{I_{1}\left( {1 - d} \right)}^{2}}}} & (8)\end{matrix}$

[0050] Hence, the input resistance r_(i) of the converter is$\begin{matrix}{r_{i} = {\frac{v_{i}}{I_{1}} = \frac{\left( {1 - d} \right)^{2}}{2C\quad f_{s}}}} & (9)\end{matrix}$

[0051] where f_(s)=1/T_(s) is the switching frequency.

[0052] Moreover, the voltage stress across the main switch S, v_(stress), equals $\begin{matrix}{v_{stress} = {{{v_{C}\left( T_{S} \right)} + V_{o}} = {{\frac{I_{1}}{C}\left( {1 - d} \right)T_{S}} = {\frac{2}{1 - d}v_{i}}}}} & (10)\end{matrix}$

[0053] In the first embodiment of the present invention, to be describedfurther below, equations (9) and (10) will be used to locate the MPP ofa solar panel. Since, as is known, the input resistance and the voltagestress across the main switch of a Cuk converter is same as (9) and(10), respectively, both SEPIC and Cuk converters exhibit similar r_(i)and v_(stress) and thus they can be used to locate the MPP.

[0054] C. Dynamic Input Resistance of the Converter under Perturbation

[0055] If a small-signal sinusoidal perturbation 8d is injected into d,

d =D+δ d =D+ {circumflex over (d)} sin ωt,  (11)

[0056] where ω=2πf and D is the nominal duty cycle at the MPP, and{circumflex over (d)} and f are the amplitude and frequency of theinjected perturbation, respectively. In the following derivations, thevalue of f is assumed to be much smaller than f_(s).

[0057] By substituting (11) into (9), the input resistance can beexpressed as $\begin{matrix}{r_{i} = {\frac{\left( {1 - D} \right)^{2}}{2f_{S}C} - {\frac{\left( {1 - D} \right)}{f_{S}C}\hat{d}\quad \sin \quad \omega \quad t} + {\frac{1}{2f_{S}C}{\hat{d}}^{2}\sin^{2}\quad \omega \quad t}}} & (12)\end{matrix}$

[0058] Hence, r_(i) includes two main components, namely the staticresistance R_(i) at the MPP and the dynamic resistance δr _(i) aroundthe MPP. Each one can be expressed as $\begin{matrix}{R_{i} = \frac{\left( {1 - D} \right)^{2}}{2f_{S}C}} & (13) \\{{\text{and}\quad \delta \quad r_{i}} = {{{- \frac{\left( {1 - D} \right)}{f_{S}C}}\hat{d}\quad \sin \quad \omega \quad t} + {\frac{1}{2\quad f_{S}C}{\hat{d}}^{2}\quad \sin^{2}\quad \omega \quad t}}} & (14)\end{matrix}$

[0059] By substituting (14) into (4), the input voltage variation δv_(i)at the MPP can be expressed as $\begin{matrix}{{{\delta \quad v_{i}} = {{{\delta \quad {\overset{\_}{v}}_{i}} + {\delta \quad {\overset{\sim}{v}}_{i}\quad \text{and}\quad \delta \quad {\overset{\sim}{v}}_{i}}} = {{\delta \quad {\overset{\sim}{v}}_{i,1}} + {\delta \quad {\overset{\sim}{v}}_{i,2}}}}}{{{\text{where}\quad \delta \quad {\overset{\sim}{v}}_{i}} = {\frac{V_{i}}{4\left( {1 - D} \right)^{2}}{\hat{d}}^{2}}},{{\delta \quad {\overset{\sim}{v}}_{i,1}} = {{- \frac{V_{i}}{\left( {1 - D} \right)}}\hat{d}\quad \sin \quad \omega \quad t}},{{\text{and}\quad \delta \quad {\overset{\sim}{v}}_{i,2}} = {{- \frac{V_{i}}{4\left( {1 - D} \right)^{2}}}{\hat{d}}^{2}\quad \cos \quad 2\omega \quad {t.}}}}} & (15)\end{matrix}$

[0060] δv_(i) is maximum when $\begin{matrix}{{{\omega \quad t} = {\frac{\left( {{2n} + 1} \right)}{2}\pi}},{n = 1},3,5,\ldots} & (16)\end{matrix}$

[0061] Its maximum value δ{overscore (V)}_(i,max) can be shown to beequal to $\begin{matrix}{{\delta \quad v_{i,\max}} = {{\frac{V_{i}}{\left( {1 - D} \right)}\hat{d}} + {\frac{V_{i}}{2\left( {1 - D} \right)^{2}}{\hat{d}}^{2}}}} & (17)\end{matrix}$

[0062] Consider the ac-component of δv_(i), its maximum valueδ{overscore (V)}_(i,max) can be expressed as $\begin{matrix}{{{\delta \quad {\overset{\sim}{v}}_{i,\max}} = {{\delta \quad {\overset{\sim}{v}}_{i,{m1}}} + {\delta \quad {\overset{\sim}{v}}_{i,{m2}}}}}{{\text{where}\quad \delta \quad {\overset{\sim}{v}}_{i,{m1}}} = {{\frac{V_{i}}{\left( {1 - D} \right)}\hat{d}\quad \text{and}\quad \delta \quad {\overset{\sim}{v}}_{i,{m2}}} = {\frac{V_{i}}{4\left( {1 - D} \right)^{2}}{{\hat{d}}^{2}.}}}}} & (18)\end{matrix}$

[0063] The ratio between the magnitude of δ{overscore (V)}_(i,1) andSδ{overscore (V)}_(i,m2),

, is $\begin{matrix}{\Re = {{\frac{\delta \quad {\overset{\sim}{v}}_{i,{m2}}}{\delta \quad {\overset{\sim}{v}}_{i,{m1}}}} = \frac{\hat{d}}{4\left( {1 - D} \right)}}} & (19)\end{matrix}$

[0064]

is an index showing the spectral quality of the input voltage variationat the frequency of the injected perturbation with respect to theamplitude of the perturbation. The smaller the value of

is, the more dominant is the component of the injected frequency inδv_(i).

[0065] D. Voltage Stress of the Main Switch Under Perturbation

[0066] The maximum value of V_(stress) (i.e., V_(stress, max)) under asinusoidal perturbation can be obtained by substituting d=D+δd andv_(i)=V_(i)+δv_(i) into (10). Thus, $\begin{matrix}{\begin{matrix}{v_{stress} = \quad {\frac{2}{\left( {1 - D - {\delta \quad d}} \right)}\left( {V_{i} + {\delta \quad v_{i}}} \right)}} \\{= \quad {\frac{2}{\left( {1 - D} \right)}\frac{1}{1 - \frac{\delta \quad d}{\left( {1 - D} \right)}}\left( {V_{i} + {\delta \quad v_{i}}} \right)}} \\{= \quad {{\frac{2}{\left( {1 - D} \right)}\left\lbrack {1 + {\frac{1}{\left( {1 - D} \right)}\delta \quad d} + {\frac{1}{\left( {1 - D} \right)^{2}}\delta \quad d^{2}} + \ldots} \right\rbrack}\left( {V_{i} + {\delta \quad v_{i}}} \right)}} \\{= \quad {{\frac{2}{\left( {1 - D} \right)}V_{i}} + {\frac{2\delta \quad d}{\left( {1 - D} \right)^{2}}\left( \frac{1}{1 - \frac{\delta \quad d}{1 - D}} \right)}}} \\{\quad {\left( {V_{i} + {\delta \quad v_{i}}} \right) + {\frac{2}{1 - D}\delta \quad v_{i}}}} \\{= \quad {V_{stress} + {\delta \quad v_{stress}}}}\end{matrix}{{\text{where}\quad V_{stress}} = \frac{2V_{i}}{\left( {1 - D} \right)}}\quad {{\text{and}\quad \delta \quad v_{stress}} = {{\frac{2\delta \quad d}{\left( {1 - D} \right)^{2}}\left( \frac{1}{1 - \frac{\delta \quad d}{1 - D}} \right)\left( {V_{i} + {\delta \quad v_{i}}} \right)} + {\frac{2}{\left( {1 - D} \right)}\delta \quad {v_{i}.}}}}} & (20)\end{matrix}$

[0067] The maximum value of v_(stress), v_(stress,max), can beapproximated by substituting δd={circumflex over (d)} andδv_(i)=δv_(i,max) in (17) into (20). It can be shown that$\begin{matrix}{{v_{{stress},\max} = {\frac{2V_{i}}{\left( {1 - D} \right)}\left\lbrack {1 + {ɛ(D)}} \right\rbrack}}{{\text{where}\quad {ɛ(D)}} = {\frac{2{\hat{d}\left( {1 - D + \frac{\hat{d}}{4}} \right)}}{\left( {1 - D} \right)\left( {1 - D - \hat{d}} \right)}.}}} & (21)\end{matrix}$

[0068] Comparing (18) and (21), it can be shown that $\begin{matrix}{{{\delta \quad {\overset{\sim}{v}}_{i,\max}} = {\beta \quad v_{{stress},\max}}},{\beta = {\frac{\hat{d}}{2}\left\lbrack \frac{\left( {1 - D - \hat{d}} \right)\left( {1 - D + \frac{\hat{d}}{4}} \right)}{\left( {1 - D} \right)^{2} + {\hat{d}\left( {1 - D + \frac{\hat{d}}{2}} \right)}} \right\rbrack}}} & (22)\end{matrix}$

[0069] at the MPP. If {circumflex over (d)}<<1−D, β≅{circumflex over(d)}/2. Thus, δ{overscore (v)} _(i,max) and V_(stress,max) form arelatively constant ratio of 13 at the MPP.

[0070]FIG. 4 is a block diagram of apparatus for locating the MPPaccording to a first embodiment of the invention. First, the erroramplifier compares the maximum input ripple voltage (i.e., δ{overscore(v)}_(i,max)) and the attenuated switch voltage stress (i.e.,β′_(stress,max)) and generates an error signal. Theoretically, β′ shouldbe equal to 1 in (22). However, as β is dependent on D, a constant valueis used to represent it for the sake of simplicity in theimplementation. Its value is equal to r₂/(r₁+r₂) so that $\begin{matrix}{\beta^{\prime} = {\frac{r_{2}}{r_{1} + r_{2}} = {\frac{1}{D_{\max} - D_{\min}}{\int_{D_{\min}}^{D_{\max}}{{\beta (D)}{D}}}}}} & (23)\end{matrix}$

[0071] where D_(min) and D_(max) are the minimum and maximum duty cycleof the main switch, respectively.

[0072] D_(max) is determined by the minimum input resistance R_(i),minof the converter, which is also the minimum equivalent output resistanceof the solar panel. By using (9), $\begin{matrix}{D_{\max} = {1 - \sqrt{2R_{i,\min}C\quad f_{S}}}} & (24)\end{matrix}$

[0073] For the converter operating in DCV mode, it must be ensured thatd₁≦d. The output current I_(o) can be expressed as $\begin{matrix}{I_{o} = {\frac{V_{o}}{R} = {\left. {{\left( {1 - d} \right)I_{1}} + {\left( {1 - d_{1}} \right)I_{2}}}\Rightarrow I_{2} \right. = {\frac{1}{1 - d_{1}}\left\lbrack {\frac{V_{o}}{R} - {\left( {1 - d} \right)I_{1}}} \right\rbrack}}}} & (25)\end{matrix}$

[0074] d₁ is determined by substituting (6) and (7) into (25) and thus$\begin{matrix}{D_{\min} = \sqrt{2\quad R\quad C\quad f_{S}}} & (26)\end{matrix}$

[0075] Next, a small-signal sinusoidal perturbation is superimposed onthe error signal and then the combined signal v _(con) is compared to aramp function to generate a PWM gate signal to the main switch.

[0076] The tracking action can be illustrated by considering the valuesof δ{overscore (v)}_(i,max) and v _(stess,max) when d does not equal D.Based on FIG. 1 and using (9), it can be shown that $\begin{matrix}{\begin{matrix}{v_{i} = \quad {\frac{r_{i}}{r_{i} + r_{g}}v_{g}}} \\{\left. \Rightarrow{\delta \quad v_{i}} \right. = \quad {{- \frac{2r_{i}r_{g}}{\left( {r_{i} + r_{g}} \right)^{2}}}\frac{v_{g}}{\left( {1 - d} \right)}\delta \quad d}}\end{matrix}\text{Thus,}} & (27) \\{{\delta \quad {\overset{\sim}{v}}_{i,\max}} = {\frac{2\alpha}{\left( {1 + \alpha} \right)^{2}}\frac{v_{g}}{\left( {1 - d} \right)}\hat{d}}} & (28)\end{matrix}$

[0077] where α=r_(i)/r_(g)=[(1−d)/(1−D)]².

[0078] By substituting (27) and (28) into (20), V_(stress,max) is equalto $\begin{matrix}{v_{{stress},\max} = {\frac{2{\alpha \left\lbrack {{\left( {1 + \alpha} \right)\left( {1 - d} \right)} + {2\hat{d}}} \right\rbrack}}{\left( {1 - d} \right)\left( {1 - d - \hat{d}} \right)\left( {1 + \alpha} \right)^{2}}v_{g}}} & (29)\end{matrix}$

[0079] Referring to (22), if {circumflex over (d)}<<1−d, β≅{circumflexover (d)}/2. It can be shown that $\begin{matrix}{\Phi = {{\frac{\beta^{\prime}v_{{stress},\max}}{\delta \quad v_{i,\max}} \cong {\frac{1}{2}\left\lbrack \frac{{\left( {1 + \alpha} \right)\left( {1 - d} \right)} + {2\hat{d}}}{1 - d - \hat{d}} \right\rbrack} \cong {\frac{1}{2}\left( {1 + \alpha} \right)}} = {\frac{1}{2}\left\lbrack {1 + \left( \frac{1 - d}{1 - D} \right)^{2}} \right\rbrack}}} & (30)\end{matrix}$

[0080] When r_(i) equals r_(g) (i.e., α=1), Φ becomes unity. This is thecondition when the converter is at the MPP. If d is smaller than D,r_(i) will be larger than r_(g) (i.e., α>1), Φ becomes larger thanunity. The error amplifier will then generate a signal so as to increasethe duty cycle. Conversely, if d is larger than D, r_(i) will be smallerthan r_(g) (i.e., α <1). Φ becomes less than unity. The error amplifierwill then generate a signal so as to decrease the duty cycle. The aboveregulatory actions cause the feedback network to adjust the duty cycle,in order to make Φ=1 or r_(i)=r_(g).

[0081] The embodiment of FIG. 4 has been experimentally checked usingthe set-up shown in FIG. 5 and using a solar panel Siemens SM-10 with arated output power of 10W. The component values of the SEPIC converterare as shown in FIG. 4. The output resistance R equals 10Ω. Theswitching frequency is set at 80 kHz and the injected sinusoidalperturbation frequency is 500 Hz. The radiation level illuminated on thesolar panel is adjusted by controlling the power of a 900W halogen lampusing a light dimmer. The bypass switch is used to give the maximumbrightness from the lamp for studying the transient response. Thesurface temperature of the panel is maintained at about 40° C. Themeasured v_(g)-i_(g) characteristics and the output power versus theterminal resistance of the solar panel at different power P_(lamp) tothe lamp are shown in FIG. 6(a) and FIG. 6(b), respectively. Under agiven P_(lamp), it can be seen that the panel output power will be atits maximum under a specific value of the terminal resistance. WhenP_(lamp) equals 900W (i.e., full power), the required terminalresistance is 14Ω, in order to extract maximum power from the solarpanel. Thus, by applying (24) and (26), D_(min) and D_(max) equal 0.274and 0.675, respectively. Based on (9), the variation of the inputresistance is between 14 Ω and 70 Ω, which are well within the requiredtracking range of the input resistance shown in FIG. 6(b).

[0082] Detailed experimental waveforms of the gate signal, the switchvoltage stress, the converter input terminal voltage, and the inputinductor current in one switching cycle at the maximum lamp power areshown in FIG. 7. Macroscopic views of the switch voltage stress, inputvoltage, and input current are shown in FIG. 8. It can be seen that alow-frequency variation of 500 Hz is superimposed on all waveforms. Theyare all in close agreement with the theoretical ones. In addition, theinput current is continuous. Thus, the MPP tracking method and apparatusof this embodiment of the present invention is better than the one usingclassical buck-type converter which takes pulsating input current.Moreover, it is unnecessary to interrupt the system, in order to testthe open-circuit terminal voltage of the solar panel.

[0083]FIG. 9 shows the ac-component of the converter input terminalvoltage with 91 equal to 0.02, 0.05, and 0.1, respectively. As

increases, the ac-component will be distorted because the second-orderharmonics become dominant in (15).

[0084] In order to observe the feedback action of the proposed approachunder a large-signal variation in the radiation level, P_(lamp) ischanged from 500W to 900W. The transient waveform of the feedback signalis shown in FIG. 10. The settling time is about 0.4 seconds. Based onthe results in FIG. 6(b), a comparison of the maximum attainable outputpower and the measured output power with the proposed control schemeunder different P_(lamp) is shown in FIG. 11. It can be seen that theproposed control technique can track the output power of the panel withan error of less than 0.2W. A major reason for the discrepancy is due tothe variation of D with respect to the duty cycle shown in (23), whichwill directly affect the tracking accuracy.

[0085] The methodology of this first embodiment of the invention isbased on connecting a pulsewidth-modulated (PWM) DC/DC converter betweena solar panel and a load or battery bus. In this embodiment a SEPICconverter operates in discontinuous capacitor voltage mode whilst itsinput current is continuous. By modulating a small-signal sinusoidalperturbation into the duty cycle of the main switch and comparing themaximum variation in the input voltage and the voltage stress of themain switch, the maximum power point (MPP) of the panel can be located.The nominal duty cycle of the main switch in the converter is adjustedto a value, so that the input resistance of the converter is equal tothe equivalent output resistance of the solar panel at the MPP. Thisapproach ensures maximum power transfer under all conditions withoutusing microprocessors for calculation.

[0086] In the first embodiment of the invention described above, a smallperturbation is introduced into the duty cycle of at least one switchingdevice in the converter. In a second embodiment of the invention, to bedescribed in more detail below, a small perturbation may be introducedinto the switching frequency of a PWM DC/DC converter. Before describingthe second embodiment in more detail, further theoretical explanation isoffered below. SEPIC and Cuk converters operating in discontinuousinductor current mode (DICM) and discontinuous capacitor voltage mode(DCVM) are illustrated.

[0087] A. Discontinuous Inductor Current Mode (DICM)

[0088] The input characteristics of SEPIC (FIG. 2) and Cuk converters(FIG. 12) are similar. The input resistance r_(i) equals $\begin{matrix}{{r_{i} = \frac{2L_{e}f_{S}}{d^{2}}},} & (31)\end{matrix}$

[0089] where L _(e) =L ₁ //L ₂ , f _(s) is the switching frequency, andd is the duty cycle of the switch S in FIGS. 2 and 12.

[0090] By differentiating (31) with respect to f_(s), it can be seenthat a small change of f_(s) will introduce a small variation in r_(i).That is, $\begin{matrix}{{\delta \quad r_{i}} = {\frac{2L_{e}}{d^{2}}\delta \quad {f_{S}.}}} & (32)\end{matrix}$

[0091] Hence, if f_(s) is modulated with a small-signal sinusoidalvariation

f _(S) ={overscore (f)} _(S)+δ{overscore (f)}_(S) ={overscore (f)} _(S)+{overscore (f)} _(S) sin(2 πf_(m) t),  (33)

[0092] where {overscore (f)} _(S) is the nominal switching frequency,f_(m) is the modulating frequency and is much lower than {overscore (f)}_(S), and {overscore (f)} _(S) is the maximum frequency deviation.

[0093] Thus, with the above switching frequency perturbation, r_(i) willinclude an average resistance R_(i) and a small variation δr_(i). Thatis,

r _(i) =R _(i) +δr _(i),  (34)

[0094] $\begin{matrix}{{{\text{where}\quad R_{i}} = {\frac{2L_{e}}{d^{2}}{\overset{\_}{f}}_{S}}},} & (35) \\{{\text{and}\quad \delta \quad r_{i}} = {\frac{2L_{e}}{d^{2}}{\hat{f}}_{S}{{\sin \left( {2\pi \quad f_{m\quad}t} \right)}.}}} & (36)\end{matrix}$

[0095] Let D_(MP) be the required duty cycle of S at MPP. r_(g) can beexpressed as $\begin{matrix}{r_{g} = {\frac{2L_{e}{\overset{\_}{f}}_{S}}{D_{MP}^{2}}.}} & (37)\end{matrix}$

[0096] By using (35) and (37), $\begin{matrix}{V_{i} = {{\frac{R_{i}}{R_{i} + r_{g}}v_{g}} = {\frac{D_{MP}^{2}}{D_{MP}^{2} + d^{2}}v_{g}}}} & (38)\end{matrix}$

[0097] and the variation of vi with respect to r_(i) becomes$\begin{matrix}{{{\delta \quad v_{i}} \approx {\frac{}{r_{i}}\left( {\frac{r_{i}}{r_{i} + r_{g}}v_{g}} \right)\delta \quad r_{i}}} = {\frac{r_{g}v_{g}}{\left( {R_{i} + r_{g}} \right)^{2}}\delta \quad r_{i}}} & (39)\end{matrix}$

[0098] By substituting (32), (35), and (37) into (39), the small-signalvariation on v_(i) is $\begin{matrix}{{\delta \quad v_{i}} = {\frac{\left( {D_{MP}d} \right)^{2}v_{g}}{\left( {D_{MP}^{2} + d^{2}} \right)^{2}{\overset{\_}{f}}_{S}}\delta \quad {f_{S}.}}} & (40)\end{matrix}$

[0099] The peak value of δv_(i) (i.e., {circumflex over (v)}_(i))becomes $\begin{matrix}{{\hat{v}}_{i} = {\frac{\left( {D_{MP}d} \right)^{2}v_{g}}{\left( {D_{MP}^{2} + d^{2}} \right)^{2}{\overset{\_}{f}}_{S}}{{\hat{f}}_{S}.}}} & (41)\end{matrix}$

[0100] As v_(g) and r_(g) vary with insolation and temperature, d shouldbe automatically adjusted to D_(MP) in the controller. The followingequation holds at the MPP and is obtained by substituting (32) and (35)into (30), $\begin{matrix}{{\frac{{\hat{f}}_{S}}{2\overset{\_}{f_{S}}}V_{i}} = {\hat{v}}_{i}} & (42)\end{matrix}$

[0101] Based on (38) and (41), the difference, ε₁, between thenormalized characteristics of$\frac{{\hat{f}}_{S}V_{i}}{2{\overset{\_}{f}}_{S}v_{g}}\quad {and}\quad \frac{{\hat{v}}_{i}}{v_{g}}$

[0102] can be shown to be equal to $\begin{matrix}{{ɛ_{1}(k)} = {{\frac{{\hat{f}}_{S}V_{i}}{2{\overset{\_}{f}}_{S}v_{g}} - \frac{{\hat{v}}_{i}}{v_{g}}} = {{\beta 1} - \frac{k^{2}}{\left( {1 + k^{2}} \right)^{2}}}}} & (43)\end{matrix}$

[0103] where k=d/D_(MP) and β={circumflex over (f)}_(S)/(2 {overscore(f)}_(S)).

[0104]FIG. 13 shows the relationships between ε₁/β and k. It can beconcluded that,

Ifd<D _(MP)(i.e., k<1), ε₁(k)>0  (44a)

If d=D _(MP)(i.e., k=1), ε₁(1)=0  (44b)

If d>D _(MP)(i.e., k>1), ε₁(k)<0  (44c)

[0105] Based on (44), the proposed MPP tracking method of a secondembodiment of the invention is shown as a block diagram in FIG. 14. f_(s) is modulated with a small-signal sinusoidal variation. Vi and{overscore (v)} _(i) are sensed. Vi is then scaled down by the factor ofβ and is compared with {circumflex over (v)} _(i) . {circumflex over(v)} _(i) is obtained by using a peak detector to extract the value ofthe ac component in v_(i). The switching frequency component in v_(i) isremoved by using a low-pass (LP) filter. The error amplifier controlsthe PWM modulator to locate d at D_(MP). If {circumflex over (v)} _(i)is smaller than ({circumflex over (f)}_(s)/2{circumflex over (f)}_(s))V_(i), ε₁>0. The output of the error amplifier, and hence d, will beincreased. Conversely, d will be decreased until d=D_(MP). It can beseen from the above than the proposed technique will keep track theoutput characteristics of solar panels without approximating thevoltage-current relationships.

[0106] B. Discontinuous Capacitor Voltage Mode (DCVM)

[0107] In this mode, r_(i) equals $\begin{matrix}{r_{i} = \frac{\left( {1 - d} \right)^{2}}{2f_{S}C}} & (45)\end{matrix}$

[0108] Thus, δr _(i) with respect to the frequency variation δf _(s) is$\begin{matrix}{{\delta \quad r_{i}} = {{- \frac{\left( {1 - d} \right)^{2}}{2{\overset{\_}{f}}_{S}^{2}C}}\delta \quad {f_{S}.}}} & (46)\end{matrix}$

[0109] Similar to deriving (38) and (40), it can be shown that$\begin{matrix}{V_{i} = {\frac{\left( {1 - d} \right)^{2}}{\left( {1 - d} \right)^{2} + \left( {1 - D_{MP}} \right)^{2}}v_{g}\quad {and}}} & (47) \\{{\hat{v}}_{i} = {\frac{\left( {1 - d} \right)^{2}\left( {1 - D_{MP}} \right)^{2}v_{g}}{\left\lbrack {\left( {1 - d} \right)^{2} + \left( {1 - D_{MP}} \right)^{2}} \right\rbrack^{2}{\overset{\_}{f}}_{S}}{\hat{f}}_{S}}} & (48)\end{matrix}$

[0110] By substituting d=D_(MP) into (37) and (48), (42) is still valid.Again, the difference, ε ₂, between the nominal characteristics of$\frac{{\hat{f}}_{S}V_{i}}{2{\overset{\_}{f}}_{S}v_{g}}\quad {and}\quad \frac{{\hat{v}}_{i}}{v_{g}}$

[0111] can be shown to be $\begin{matrix}{{ɛ_{2}\left( k^{\prime} \right)} = {{\frac{{\hat{f}}_{S}V_{i}}{2{\overset{\_}{f}}_{S}v_{g}} - \frac{{\hat{v}}_{i}}{v_{g}}} = \frac{\beta \quad {k^{\prime 2}\left( {k^{{\prime 2}} - 1} \right)}}{\left( {k^{\prime 2} + 1} \right)^{2}}}} & (49)\end{matrix}$

[0112] where k′=(1−d)/(1−D_(MP)).

[0113]FIG. 15 shows the relationships between ε ₂/β and k′. Similarbehaviors as in (44) are obtained

If d<D _(MP) (k′>1), ε₂(k′)>0  (50a)

If d=D _(MP) (k′=1), ε₂ (1)=0  (50b)

If d>D _(MP) (k′<1), ε₂(k′)<0  (50c)

[0114] Hence, the control method used when the converter is operated inDICM can also be applied to a converter operated in DCVM.

[0115] C. Comparison of DICM and DCVM

[0116] Although a converter operating in DICM and DCVM can perform theMPP tracking in accordance with this embodiment of the invention,selection of a suitable operating mode is based on several extrinsic andintrinsic characteristics. Table I shows a comparison of the converterbehaviors in DICM and DCVM. TABLE I Comparisons of the converterbehaviors in DICM and DCVM DICM DCVM M $\frac{d}{d_{1}}$

$\frac{d_{1}}{1 - d}$

r_(i) $\frac{2L_{e}f_{S}}{d^{2}}$

$\frac{\left( {1 - d} \right)^{2}}{2f_{S}C}$

ΔI₁ $\frac{2L_{2}}{d\left( {L_{1} + L_{2}} \right)}I_{1}$

$\begin{matrix}{{Negligible}\quad {as}} \\{L_{1}\operatorname{>>}\frac{1}{\left( {2\pi \quad f_{S}} \right)^{2}C}}\end{matrix}\quad$

V_(s, max) and V_(D,,max) (1 + M)V_(i) $\frac{2M}{d_{1}}V_{i}$

I_(s,max) and I_(D,max) $\frac{2}{{Md}_{1}}I_{1}$

$\left( {1 + \frac{1}{M}} \right)I_{1}$

d₁ $\sqrt{2L_{e}{f_{S}/R}}$

$\sqrt{2{Rf}_{S}C}$

Condition of d <1 − d₁ >d₁ Application High voltage, Low voltage, lowcurrent high current Recommended arrangement Series connection Parallelconnection for solar panels

[0117] For the extrinsic characteristics, apart from the difference inthe voltage conversion ratio M, the input current ripple ΔI ₁ in theDCVM is smaller than that in the DJCM. Thus, variation of thepanel-converter operating point in the DCVM is smaller. This caneffectively operate the panel at the near MPP. Nevertheless, inputcurrent perturbation is designed to be less than 10% in theimplementation.

[0118] In order to ensure that the converter is operating in the DICM,$\begin{matrix}{{d < {1 - \sqrt{\frac{2L_{e}f_{s}}{R}}}} = \frac{V_{o}}{V_{o} + V_{i}}} & (51)\end{matrix}$

[0119] Thus, (51) gives the maximum duty cycle of S for a given loadresistance.

[0120] In order to ensure that the converter is operating in DCVM,$\begin{matrix}{{d > \sqrt{2R\quad f_{s}C}} = \frac{V_{o}}{V_{o} + V_{i}}} & (52)\end{matrix}$

[0121] (52) gives the minimum duty cycle of S for a given loadresistance.

[0122] For the intrinsic characteristics, the voltage stress V_(s,max)of S in the DCVM is higher than that in the DICM under the same panelterminal voltage and voltage conversion ratio. Conversely, the currentstress I_(S,max) in the DICM is higher than that in the DCVM with thesame panel output current. Thus, for the same panel power, DICM is moresuitable for panel in series connection whilst DCVM is for parallelconnection.

[0123] This second embodiment of the invention may be verified by meansof the experiment setup shown in FIG. 16. A solar panel Siemens SM-10with a rated output power of 10W is used. Two SEPICs, which areoperating in DICM and DCVM, respectively, have been prototyped. Thecomponent values of the two converters are tabulated in Table II.

[0124] Table II Component values of the two converters DICM DCVM L₁ 2.2mH 2.2 mH L₂ 25 μH 450 μH C 100 μF 47 nF C₀ 1 mF 1 mF R 10 Ω 10 Ω{overscore (ƒ)} _(s) 50 kHz 50 kHz {circumflex over (ƒ)}_(s) 10 kHz 10kHz ƒ_(m) 1 kHz 1 kHz

[0125] The switching frequency is 50 kHz. The modulating frequency f_(m)is 1 kHz. The maximum frequency deviation {circumflex over (f)}_(s) is10 kHz. Based on Table I and (31), the maximum value of d is 0.5 for theconverter in DICM. The minimum panel output resistance that can bematched by the converter is 9.8 Ω. For the converter in DCVM, based onTable I and (35), the minimum value of d is 0.217. The maximum paneloutput resistance that can be matched is 130.5 Ω. The surfacetemperature of the panel is kept at about 40° C. throughout the test.The radiation illuminated is adjusted by controlling the power of a 900Wtungsten halogen lamp using a programmable dc supply source—Kikusui PCR2000L. FIG. 17 shows the P_(o)-r_(i) characteristics of the solar panelat different P_(lamp). It can be seen that the output resistance of thepanel at MPP varies from 18 Ω to 58 Ω when P_(lamp) is changed from 900Wto 400W. The operating range is within the tracking capacity (i.e., theinput resistance) of the two converters. FIG. 18 shows the experimentalwaveforms of v_(i) and i₁ of the two prototypes at the MPP when P_(lamp)equals 900W. It can be seen that vi has a small sinusoidal perturbationof 1 kHz. FIG. 19 shows the experimental voltage and current stresses onS and D in the two converters. As expected, the current stresses on Sand D in the DICM are about three times higher than that in the DCVM,whilst the voltage stresses on S and D in the DCVM are four times higherthan that in the DICM. These confirm the theoretical prediction.

[0126] An insolation change is simulated by suddenly changing P_(lamp)from 400W to 900W. The transient waveforms of v_(i) and i_(i) of the twoconverters are given in FIG. 20. It was found that both converters canperform the MPP tracking function and the panel output power isincreased from 2.5W to 9.5W in 0.3 sec in both cases. The tracked poweris in close agreement with the measurements in FIG. 17.

[0127] It will thus be seen that at least in preferred forms of theinvention novel techniques are provided for tracking the MPP of a solarpanel in varying conditions. Both embodiments use either a PWM dc/dcconverter, for example a SEPIC or Cuk converter. In a first embodimentof the invention a small perturbation is introduced into the duty cycleof the converter operating in discontinuous capacitor voltage mode. Inthe second embodiment of the invention a PWM dc/dc converter operatingin discontinuous inductor-current or capacitor-voltage mode is used tomatch with the output resistance of the panel. In this second embodimentof the invention a small sinusoidal variation is injected into theswitching frequency and comparing the maximum variation and the averagevalue at the input voltage, the MPP can be located. Both embodiments aresimple and elegant without requiring any digital computation andapproximation of the panel characteristics.

1. A method for tracking the maximum power point of a solar panel, comprising: (a) providing a pulsewidth modulated (PWM) DC/DC converter between the output of said panel and a load, and (b) introducing a perturbation into a switching parameter of said converter.
 2. A method as claimed in claim 1 wherein said parameter is the duty cycle of at least one switching device in the converter.
 3. A method as claimed in claim 1 wherein said parameter is the switching frequency of at least one switching device in the converter.
 4. Apparatus for tracking the maximum power point of a solar panel, comprising: (a) a PWM DC/DC converter between the output of the solar panel and a load, and (b) means for introducing a perturbation into a switching parameter of said converter.
 5. Apparatus as claimed in claim 4 wherein said converter operates in switching mode and said perturbation means comprises means for introducing a perturbation into the duty cycle of at least one switching device of said converter.
 6. Apparatus as claimed in claim 4 wherein said converter operates in switching mode and said perturbation means comprises means for introducing a perturbation into the switching frequency of at least one switching device of said converter.
 7. Apparatus as claimed in claim 4 wherein said converter is a SEPIC or Cuk converter. 